5.17. among_diff_0

 DESCRIPTION LINKS GRAPH AUTOMATON
Origin

Used in the automaton of $\mathrm{\pi \pi \pi \pi \pi \pi }$.

Constraint

$\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathtt{0}\left(\mathrm{\pi ½\pi  \pi °\pi },\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$

Arguments
 $\mathrm{\pi ½\pi  \pi °\pi }$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $\mathrm{\pi ½\pi  \pi °\pi }\beta ₯0$ $\mathrm{\pi ½\pi  \pi °\pi }\beta €|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi \pi \pi }\right)$
Purpose

$\mathrm{\pi ½\pi  \pi °\pi }$ is the number of variables of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ that take a value different from 0.

Example
$\left(3,β©0,5,5,0,1βͺ\right)$

The $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathtt{0}$ constraint holds since exactly 3 values of the collection of values $\beta ©0,5,5,0,1\beta ͺ$ are different from 0.

Typical
 $\mathrm{\pi ½\pi  \pi °\pi }>0$ $\mathrm{\pi ½\pi  \pi °\pi }<|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>1$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$$\left(1,\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },0\right)$
Symmetries
• Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ are permutable.

• An occurrence of a value of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }$ that is different from 0 can be replaced by any other value that is also different from 0.

See also

generalisation: $\mathrm{\pi \pi \pi \pi \pi }$Β ( replaced by $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }\beta \mathrm{\pi \pi \pi \pi \pi \pi }$).

Keywords
Arc input(s)

$\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$

Arc generator
$\mathrm{\pi \pi Έ\pi Ώ\pi Ή}$$\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\right)$

Arc arity
Arc constraint(s)
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=\mathrm{\pi ½\pi  \pi °\pi }$

Graph model

Since this is a unary constraint we employ the $\mathrm{\pi \pi Έ\pi Ώ\pi Ή}$ arc generator in order to produce an initial graph with a single loop on each vertex.

PartsΒ (A) andΒ (B) of FigureΒ 5.17.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi }$ graph property, the loops of the final graph are stressed in bold.

Automaton

FigureΒ 5.17.2 depicts the automaton associated with the $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathtt{0}$ constraint. To each variable ${\mathrm{\pi  \pi °\pi }}_{i}$ of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ corresponds a 0-1 signature variable ${\mathrm{\pi }}_{i}$. The following signature constraint links ${\mathrm{\pi  \pi °\pi }}_{i}$ and ${\mathrm{\pi }}_{i}$: . The automaton counts the number of variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection that take a value different from 0 and finally assigns this number to $\mathrm{\pi ½\pi  \pi °\pi }$.